2019 AMC 10B Problems/Problem 9
Contents
Problem
The function is defined by for all real numbers , where denotes the greatest integer less than or equal to the real number . What is the range of ?
Solution
There are 4 cases we need to test here:
Case 1: x is a positive integer. WLOG, assume x=1. Then f(1) = 1 - 1 = .
Case 2: x is a positive fraction. WLOG, assume x=0.5. Then f(0.5) = 0 - 0 = .
Case 3: x is a negative integer. WLOG, assume x=-1. Then f(-1) = 1 - 1 = .
Case 4: x is a negative fraction. WLOG, assume x=-0.5. Then f(-0.5) = 0 - 1 = .
Thus the range of function f is
Solution 2
It is easily verified that when is an integer, then is zero. We need only to consider the case when is not.
When is a positive number, , so
When is a negative number, let be composed of integer part and decimal part (both ):
Thus, the range of f is
Note: One could solve the case of as a negative non-integer this way:
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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